Can is a modal with multiple uses including: ability, permission, request, offer, suggestion, possibility

Ability: We can speak Spanish

Permission: My mom said I can go out on Friday night

Request: Can you pick up some milk on your way home?

Offer: I can bake Linda’s birthday cake if you don’t have time.

Suggestion: We can go to the amusement park next weekend

Possibility: Exams can be really stressful

May can be used to express possibility, permission

Possibility: I may go out tonight

Permission: You may proceed to the front gate

Must/Mustn’t used to express obligation

Obligation: You must study for the biology exam.

Should/Shouldn’t used for advice and obligation

Advice: You should wear a warm jacket, it’s really cold out

Obligation: You should call your mother

Would used for requests and habits

Request: Would you be able to work on Saturday?

Habits (past): I would usually go dancing on Wednesdays

Could used for ability, polite request, offer, suggestion, possibility 

Ability: I could ride a unicycle when I was a kid.

Polite request: Could I have a glass of wine, please?

Offer: Could I interest you in a glass of wine?

Suggestion: You could catch the ten o’clock bus tomorrow.

Possibility: I could pick you up from soccer practice.

Might is used for possibility

Possibility: I might travel to Asia next year.

Ought to used for advice 

Advice: You ought to check the papers for job listings

Shall used for suggestion

Suggestion: We shall see what happens.

Will used for prediction, promise, habits, affirmation, request 

Prediction: I think the fundraiser will be very successful.

Promise: I will be there at 7 o’clock on the dot.

Affirmation: “Drive carefully” “I will!”

Request: Will you join my for lunch this afternoon?